Probability

We will calculate each of the probabilities step by step using combinations and the structure of a standard deck of 52 playing cards.

Total possible 5-card hands:

The total number of ways to choose 5 cards out of 52 is:

(525)=2, ⁣598, ⁣960\binom{52}{5} = 2,\!598,\!960(552​)=2,598,960

(a) Probability of a Straight

A straight is 5 cards in sequence (e.g. A-2-3-4-5, 2-3-4-5-6, ..., 10-J-Q-K-A), not all the same suit (if they are, it's a straight flush). We ignore suit and order.

Step 1: Count possible ranks for a straight

There are 10 possible sequences of 5 consecutive ranks (starting with A-2-3-4-5 up to 10-J-Q-K-A).

Step 2: For each sequence, count suit combinations

Each card in the 5-card sequence can be of any of the 4 suits.

So number of ways to assign suits to the 5 cards:

45=10244^5 = 102445=1024

But we exclude straight flushes, where all 5 cards are of the same suit:
There are 4 straight flushes for each of the 10 sequences, so:

10×4=40 straight flushes10 \times 4 = 40 \text{ straight flushes}10×4=40 straight flushes

So total number of non-straight-flush straights:

10×(45−4)=10×(1024−4)=10×1020=10, ⁣20010 \times (4^5 - 4) = 10 \times (1024 - 4) = 10 \times 1020 = 10,\!20010×(45−4)=10×(1024−4)=10×1020=10,200

Final Probability:

\frac{10,\!200}{2,\!598,\!960} \approx 0.0039 \quad \text{(or about 0.39%)}

(b) Probability of a Flush

A flush is 5 cards of the same suit, not in sequence.

Step 1: Choose a suit

There are 4 suits.

Step 2: Choose any 5 cards from the 13 cards in that suit:

(135)=1287\binom{13}{5} = 1287(513​)=1287

So total flushes (including straight flushes):

4×1287=51484 \times 1287 = 51484×1287=5148

Now subtract the 40 straight flushes (from part a):

5148−40=51085148 - 40 = 51085148−40=5108

Final Probability:

\frac{5108}{2,\!598,\!960} \approx 0.001965 \quad \text{(or about 0.197%)}

(c) Probability of a Full House

A full house is 3 cards of one rank and 2 cards of another rank.

Step 1: Choose rank for the triplet

There are 13 ranks, choose 1:

(131)=13\binom{13}{1} = 13(113​)=13

Choose 3 cards of that rank from 4 suits:

(43)=4\binom{4}{3} = 4(34​)=4

Step 2: Choose rank for the pair (different from triplet)

There are 12 remaining ranks:

(121)=12\binom{12}{1} = 12(112​)=12

Choose 2 cards of that rank:

(42)=6\binom{4}{2} = 6(24​)=6

Total number of full houses:

13×4×12×6=374413 \times 4 \times 12 \times 6 = 374413×4×12×6=3744

Final Probability:

\frac{3744}{2,\!598,\!960} \approx 0.001440 \quad \text{(or about 0.144%)}

Final Answers:

(a) ~0.0039 (0.39%)
(b) ~0.001965 (0.197%)
(c) ~0.001440 (0.144%)